At the final analysis at time \(t_{end}\)  at which the outcome
has been ascertained on all \(n_{max}\)  subjects,
\(\widehat{\beta }_{\mathcal {T}_F}(t_{end})\)  and \(\widehat{\beta }_{IPW}(t_{end})\)  yield
(versions of) the intended full data analysis; and
\(\widehat{\beta }_{AIPW1}(t_{end})\)  and \(\widehat{\beta }_{AIPW2}(t_{end})\)  are
identical and yield the covariate-adjusted analysis exploiting
baseline covariate information discussed at the end of
Section REF . In all scenarios, for the null
hypothesis \(\mbox{H}_0\!\!: \beta _0 = 0\)  and
one-sided alternative hypotheses and level of significance
\(\alpha = 0.025\) , we used the R package ldbounds[1]} with
a Lan-DeMets spending function to compute both
O'Brien-Fleming[2]} and Pocock[3]} stopping
boundaries at each analysis time \(t_j\) , \(j=1,\ldots ,K+1\) . For
\(\widehat{\beta }_{\mathcal {T}_F}(t_j)\) , this calculation was based on the proportion
of information \(p(t_j) = n_A(t_j)/n_{max}\) ; for each of the IPWCC
and AIPWCC estimators \(\widehat{\beta }_{IPW}(t_j)\) ,
\(\widehat{\beta }_{AIPW1}(t_j)\) , and \(\widehat{\beta }_{AIPW2}(t_j)\) , the stopping
boundaries were obtained using the approximate proportion of
information (REF ) based on the relevant
approximate effective sample size \(n_{ESS}(t_j)\)  given in (REF ).
